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Incorporate Jonathans's corrections to Conclusion

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Andreas Tsouchlos
2026-05-04 15:28:11 +02:00
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src/thesis/chapters/5_conclusion_and_outlook.tex
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% Recap of motivation % Recap of motivation
This thesis investigated decoding under \acp{dem} for fault-tolerant This thesis investigates decoding under \acp{dem} for fault-tolerant
\ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes. \ac{qec}, with a focus on low-latency decoding methods for \ac{qldpc} codes.
The repetition of the syndrome measurements, especially under The repetition of the syndrome measurements, especially under
consideration of circuit-level noise, leads to a significant increase consideration of circuit-level noise, leads to a significant increase
in decoding complexity: in our experiments on the $\llbracket in decoding complexity: in our experiments on the $\llbracket
144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction 144,12,12 \rrbracket$ \ac{bb} code with $12$ syndrome extraction
rounds, the check matrix grew from 144 \acp{vn} and 72 rounds, the check matrix grows from 144 \acp{vn} and 72
\acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}. \acp{cn} to 9504 \acp{vn} and 1008 \acp{cn}.
% Recap of research gap and own work % Recap of research gap and own work
Sliding-window decoding addresses the latency constraint by Sliding-window decoding addresses the latency constraint by
exploiting the time-like locality of the syndrome extraction circuit, exploiting the time-like locality of the syndrome extraction circuit.
which manifests as a block-diagonal structure in the detector error This manifests as a block-diagonal structure in the detector error
matrix when detectors are defined as the difference of consecutive matrix when detectors are defined as the difference of consecutive
syndrome measurement rounds. syndrome measurement rounds.
We drew a comparison to windowed decoding for \ac{sc}-\ac{ldpc} We draw a comparison to windowed decoding for \ac{sc}-\ac{ldpc}
codes, but noted that the existing realizations of sliding-window codes, but noted that the existing realizations of sliding-window
decoding discard the soft information produced inside one window decoding discard the soft information produced inside one window
before moving to the next. before moving to the next.
@@ -29,25 +29,26 @@ the overlap region of the previous window are reused to initialise
the corresponding messages of the next window in place of the the corresponding messages of the next window in place of the
standard cold-start initialisation. standard cold-start initialisation.
We formulated the warm start first for plain \ac{bp} and then for We formulate the warm start for standard \ac{bp} and for
\ac{bpgd}, the latter being attractive as an inner decoder because it \ac{bpgd}.
In particular the latter being attractive as an inner decoder because it
addresses the convergence problems caused by short cycles and addresses the convergence problems caused by short cycles and
degeneracy in \ac{qldpc} Tanner graphs. degeneracy in \ac{qldpc} Tanner graphs.
The decoders were evaluated by Monte Carlo simulation on the The decoders are evaluated by conducting Monte Carlo simulations on the
$\llbracket 144,12,12 \rrbracket$ \ac{bb} code over $12$ syndrome $\llbracket 144,12,12 \rrbracket$ \ac{bb} code over $12$ syndrome
extraction rounds under standard circuit-based depolarizing noise. extraction rounds under standard circuit-based depolarizing noise.
We focused on a qualitative analysis, refraining from further We focus on a qualitative analysis, refraining from further
optimizations such as introducing a normalization parameter for the optimizations such as introducing a normalization parameter for the
min-sum algorithm. min-sum algorithm.
% Recap of experimental conclusions % Recap of experimental conclusions
For plain min-sum \ac{bp}, the warm start was consistently beneficial For standard min-sum \ac{bp}, the warm start is consistently
across the parameter ranges we examined. The size of the gain depended beneficial to the cold start, across the considered parameter ranges.
on the overlap between consecutive windows: enlarging $W$ or The size of the gain depends on the overlap between consecutive
shrinking $F$, both of which enlarge the overlap, raised the windows: enlarging $W$ or shrinking $F$, both of which enlarge the
warm-start performance increase. overlap, result in larger gains of the warm-start.
We argued that the underlying mechanism is an effective increase in We observe that the underlying mechanism is an effective increase in
the number of \ac{bp} iterations spent on the \acp{vn} in the overlap the number of \ac{bp} iterations spent on the \acp{vn} in the overlap
region: each such \ac{vn} is processed by multiple consecutive window region: each such \ac{vn} is processed by multiple consecutive window
invocations, and the warm start lets these invocations accumulate invocations, and the warm start lets these invocations accumulate
@@ -55,7 +56,7 @@ iterations on the same \acp{vn} rather than restarting from scratch.
The gain was most pronounced at low numbers of maximum iterations, where The gain was most pronounced at low numbers of maximum iterations, where
every additional iteration carries proportionally more information. every additional iteration carries proportionally more information.
For \ac{bpgd}, we noted that more information is available in the For \ac{bpgd}, we note that more information is available in the
overlap region of a window: in addition to the \ac{bp} messages, overlap region of a window: in addition to the \ac{bp} messages,
there is information about which \acp{vn} were decimated and to what value. there is information about which \acp{vn} were decimated and to what value.
Passing this decimation information to the next window in addition to Passing this decimation information to the next window in addition to
@@ -65,14 +66,14 @@ overlap region.
Restricting the warm start to the \ac{bp} messages alone, removed this effect. Restricting the warm start to the \ac{bp} messages alone, removed this effect.
The resulting message-only warm start recovered a consistent The resulting message-only warm start recovered a consistent
improvement over cold-start that followed the same qualitative improvement over cold-start that followed the same qualitative
behaviour as for plain \ac{bp}: larger overlap, achieved by larger behaviour as for standard \ac{bp}: larger overlap, achieved by larger
$W$ or smaller $F$, yielded a larger gain, and the $W$ or smaller $F$, yielded a larger gain, and the
performance difference was most pronounced at low numbers of maximum iterations. performance difference was most pronounced at low numbers of maximum iterations.
% Implications from experimental results % Implications from experimental results
These observations imply that the warm-start modification to These observations imply that the warm-start modification to
sliding-window decoding provides a consistent improvement, as long as sliding-window decoding can provide a consistent improvement, as long as
some care is taken with specifying the information to be passed to some care is taken with specifying the information to be passed to
the subsequent window. the subsequent window.
Note that this comes at no additional cost to the decoding complexity, Note that this comes at no additional cost to the decoding complexity,
@@ -94,25 +95,10 @@ underlying mechanism is structural rather than code-specific, but
quantifying the gain across code families and noise models is left to quantifying the gain across code families and noise models is left to
future work. future work.
A second direction is a systematic study of inner decoders under the A second direction is a systematic study of other inner decoders under the
warm-start framework. warm-start framework, such as automorphism ensemble decoding
We considered plain min-sum \ac{bp} and \ac{bpgd}, but other \cite{koutsioumpas_automorphism_2025} or neural \ac{bp}
algorithms used for \ac{qldpc} decoding, such as automorphism \cite{miao_quaternary_2025}.
ensemble decoding \cite{koutsioumpas_automorphism_2025} or neural
\ac{bp} \cite{miao_quaternary_2025} may admit warm-start variants of their own.
A third direction is a softer treatment of the decimation state in \ac{bpgd}.
Rather than discarding the decimation information of the previous
window entirely, as in the message-only warm start used here, one
could encode the decimation decisions as strong but finite biases on
the channel \acp{llr} of the next window, allowing the new window's parity
checks to override them if the syndrome calls for it.
This would interpolate between the two warm-start variants studied here and
might combine the benefits of both.
A related question is whether the decimation schedule itself should
be aware of the window structure, for instance by deferring
decimation of \acp{vn} in the overlap region until they have been
visited by the next window.
A final direction is suggested by the structural similarity between A final direction is suggested by the structural similarity between
sliding-window decoding for \acp{dem} and windowed decoding for sliding-window decoding for \acp{dem} and windowed decoding for